Group theoretical methods in machine learning

This thesis explores applications of non-commutative harmonic analysis to machine learning. In particular, we address learning on groups and learning in a way that is invariant to a group. The former arises in learning rankings and matchings, in both cases, the group being the symmetric group. The application we explore in detail is the identity management problem in multi-object tracking. The latter arises in many contexts, none more immediate than invariance to translation, rotation, scaling, etc., in computer vision. In this domain we derive a new system of translation and rotation invariant features for images, and introduce a new system of invariants applicable to any compact group, which we call the skew spectrum. The ideas behind harmonic analysis (commutative or non-commutative) run very deep, and one of the main messages of the thesis is that algebra can unravel structure in data which might otherwise be overlooked. All this would not be of much use to practitioners, however, were it not for a family of recently developed algorithms, chief amongst them fast Fourier transforms, which are finally starting to make it possible to scale up algebraic methods to the size of real-world problems. Much of our work revolves around carefully tailoring fast Fourier transforms to solve specialized tasks demanded by applications. The thesis includes background material in both machine learning and algebra, so it can serve as a primer for researchers familiar with one field but not the other.